Functional ANOVA, ultramodularity and monotonicity: Applications in multiattribute utility theory

European Journal of Operational Research 210 (2011) 326–335

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Functional ANOVA, ultramodularity and monotonicity: Applications in multiattribute utility theory F. Beccacece, E. Borgonovo ⇑ Department of Decision Sciences and ELEUSI Research Center, Bocconi University, Via Roentgen 1, 20135 Milano, Italy

a r t i c l e

i n f o

Article history: Received 7 December 2009 Accepted 30 August 2010 Available online 26 September 2010 Keywords: Multiattribute utility theory Functional ANOVA Multi-criteria analysis Ultramodular functions

a b s t r a c t Utility function properties as monotonicity and concavity play a fundamental role in reflecting a decisionmaker’s preference structure. These properties are usually characterized via partial derivatives. However, elicitation methods do not necessarily lead to twice-differentiable utility functions. Furthermore, while in a single-attribute context concavity fully reflects risk aversion, in multiattribute problems such correspondence is not one-to-one. We show that Tsetlin and Winkler’s multivariate risk attitudes imply ultramodularity of the utility function. We demonstrate that geometric properties of a multivariate utility function can be successfully studied by utilizing an integral function expansion (functional ANOVA). The necessary and sufficient conditions under which monotonicity and/or ultramodularity of singleattribute functions imply the monotonicity and/or ultramodularity of the corresponding multiattribute function under additive, preferential and mutual utility independence are then established without reliance on the utility function differentiability. We also investigate the relationship between the presence of interactions among the attributes of a multiattribute utility function and the decisionmaker’s multivariate risk attitudes.  2010 Elsevier B.V. All rights reserved.

1. Introduction The solution of several decision-making problems requires the quantitative assessment of multiattribute objective (utility) functions [u(x)]. Practical advantages are registered in all those applications in which it is possible to elicit single-attribute functions and next aggregate them. Utility, additive and preferential independence are often assumed in the practice, since they lead to additive and multiplicative forms of the utility functions (Fishburn, 1980; Keeney and Raiffa, 1993; Keeney, 2006; Baucells et al., 2008). The elicitation methods described in Fortemps et al. (2008), Greco et al. (2008), Figueira et al. (2009), Anginella et al. (2004, 2010) play a central role for obtaining quantitative results. Monotonicity and convexity properties on the single-attribute functions are required for the resulting multiattribute utility function to be in one-to-one correspondence with the decision-maker’s preference structure and risk aversion attitudes. In the singleattribute case, non-satiation leads to monotonically increasing utility functions; risk aversion combined with non-satiation implies utility function concavity. In multiattribute utility problems, non-satiation still leads to monotonicity, while the notion of risk aversion becomes more general, with several possible

⇑ Corresponding author. Tel.: +39 3338484678. E-mail address: [email protected] (E. Borgonovo). 0377-2217/$ - see front matter  2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.08.032

extensions. Richard (1975)’s multivariate risk aversion leads to the supermodularity of u(x) (Ortega and Escudero, 2010) while, as we are to see, Tsetlin and Winkler (2009) multivariate ‘‘preferences for combining good with bad lead to the ultramodularity” of u(x). The question is, then, whether the motoniticity (ultramodularity) of u(x) implies the motoniticity (ultramodularity) of the single-attribute utility functions under the various forms of preference structures. Conversely, one would need to know whether, by eliciting single-attribute utility functions which possess a given property, She is insured that such property is maintained in u(x). This work introduces an approach for linking analytic and geometric properties of multiattribute objective functions to the properties of their single-attribute constituents, without restrictions on the preference structure. Ideally, monotonicity, concavity and ultramodularity can all be characterized in terms of partial derivatives (first and second order, respectively). However, methods for multiple criteria ranking (Greco et al., 2008; Figueira et al., 2009; Jacquet-Lagreze et al., 1987) build a set of additive value functions compatible with the revealed preference information, which are piecewise defined (mostly piecewise linear). Hence, a mathematical approach based on differentiation does not possess the required generality. We therefore shift the background to the integral expansion of a multivariate function (f) generated by the high dimensional model representation (HDMR) theory (functional ANOVA) (Efron and Stein, 1981; Rabitz and Alis, 1999; Alis and Rabitz, 2001; Sobol, 2001,

F. Beccacece, E. Borgonovo / European Journal of Operational Research 210 (2011) 326–335

2003). For this expansion to hold, in fact, the sole measurability of f is required. Functional ANOVA is a fundamental tool in statistics and global sensitivity analysis (Rabitz and Alis, 1999; Wang, 2006; Sobol, 2003). However, in spite of its widespread utilization, a study of its monotonicity and ultramodularity properties has not been offered yet. We obtain necessary and sufficient conditions that insure that, given the monotonicity and ultramodularity of f, these properties are preserved at the various orders of the expansion. We then specialize these general results to the case of additive and multiplicative functions, since these two functional forms frequently appear in MAUT. In particular, we obtain the following results: I) Under additive utility independence: (i) u is monotonic if and only if all the corresponding single-attribute conditional utility functions are; and (ii) u is ultramodular if and only if all the conditional utilities are; (II) under mutual utility independence, if and only if u is monotonic and ultramodular, then all single-attribute utility functions are; (III) letting v denote the multiattribute value function for preferences under certainty (on the distinction between u and v, we refer to Keeney and Raiffa (1993)), it is possible to prove that, under preferential independence: (i) if v is non-decreasing, then u is; (ii) if v is ultramodular, then u is. Each of the previous results is also characterized in terms of a decision-maker’s multivariate risk attitudes. In particular, it is obtained that u is ultramodular if and only if it reflects the preferences for combining good with bad stated in Tsetlin and Winkler (2009). The presence of interactions in the functional ANOVA representation of u is also investigated in its connection with multivariate risk attitudes. We show that the absence of interactions in u implies multivariate risk neutrality in the sense of Richard (1975). The remainder of this work is organized as follows. Section 2 discusses the MAUT implications of ultramodularity. Section 3 provides a review of functional ANOVA and HDMR. Section 4 discusses the monotonicity properties of the functional ANOVA expansion. Section 5 derives results for ultramodularity. Both Sections 4 and 5 apply the findings to separable functions.Section 6 discusses implications of the above findings in MAUT. Conclusions are offered in Section 7.

2. Ultramodularity in MAUT: Multivariate risk aversion The mathematical properties of a multiattribute utility function must be in one-to-one correspondence with the decision-maker’s preference structure. For instance, monotonicity reflects the nonsatiation property, which states that ‘‘more of an attribute is preferred to less” (Tsetlin and Winkler, 2009, p. 1944; Ingersoll, 1987). Such correspondence is maintained both in single-attribute and in multiattribute problems. In single-attribute problems, given the monotonicity of the utility function, concavity reflects risk aversion; if, in addition, the utility function is regular, risk aversion is univocally characterized by the Arrow–Pratt measure. In multiattribute problems, the one-to-one correspondence between concavity and risk aversion is lost and generalizations of concavity come into play. For instance, multivariate risk aversion as introduced in Richard (1975) implies the submodularity of the utility function (see Ortega and Escudero, 2010). In this section, we investigate what types of multivariate risk attitudes are linked to ultramodularity. As discussed in Marinacci and Montrucchio (2005), two alternative definitions of ultramodular functions can be stated, the first one based on the notion of test quadruple, the second one based on increasing differences. We utilize the latter definition, because it is better suited to the purposes of this paper. At the same time, we also introduce the notion of neg-ultramodular functions.

327

Definition 1. A function f : X # Rn ! R is (a) ultramodular, if

f ðx þ hÞ  f ðxÞ 6 f ðy þ hÞ  f ðyÞ;

ð1Þ

(b) neg-ultramodular, if

f ðx þ hÞ  f ðxÞ P f ðy þ hÞ  f ðyÞ

ð2Þ

for all x, y 2 X and h P 0 with x 6 y and x + h, y + h 2 X. By Definition 1, one notes that, if f is ultramodular, then f is neg-ultramodular. Definition 1 shows that ultramodular (negultramodular) functions possess the property of increasing (decreasing) differences (see Marinacci and Montrucchio, 2005, p. 315). Ultramodular functions are also known as directionally convex functions (Ortega and Escudero, 2010). Marinacci and Montrucchio (2005) highlight that ultramodularity is a generalization of scalar convexity. In fact, if and only if f : R ! R is convex, then it is ultramodular. However, ultramodularity and convexity become distinct notions for multivariate functions (Marinacci and Montrucchio, 2005, 2008). In Marinacci and Montrucchio (2005), several properties of ultramodular functions are proven. The ones relevant to this work are collected in the following proposition. Proposition 1. Let f ; g : X # Rn ! R two ultramodular functions. 1. The sum af + b g is ultramodular if aand b are non-negative scalars. 2. The sum f + k, k 2 R is ultramodular. 3. The product f  g is ultramodular if, in addition, f and g are both non-negative and increasing. 4. The composition h  f is ultramodular, provided f is monotonic and ultramodular, and h is ultramodular and increasing. Interpretations of ultramodularity have been offered in Economics and Game Theory. Marinacci and Montrucchio (2005) illustrate that ultramodularity ‘‘reflects a stronger form of complementary than supermodularity” (Marinacci and Montrucchio, 2005, p. 311). Indeed, ultramodular functions are also supermodular while the converse is not true (see Topkis, 1995; Milgrom and Shannon, 1994). Supermodularity and ultramodularity are also relevant in the theory of pseudo-Boolean functions (see Foldes and Hammer, 2005). However, a rigorous investigation of the implications of ultramodularity in MAUT has not been offered yet. The conditions that make ultramodularity appear in multiattribute problems are discussed next. Ultramodularity is connected with multivariate risk attitudes. Consider a two attribute (x, y) problem for simplicity. The main difference between single-attribute and multiattribute problems is stated in Richard (1975): ‘‘a decision-maker can be risk averse for gambles on x alone or for gambles on y alone, but still be multivariate risk seeking.” Richard (1975) then introduces the concept of multivariate risk aversion as follows. A decision-maker is multivariate risk averse if She ‘‘prefers getting some of the best and some of the worst to taking a chance on all of the best or all of the worst.” Richard (1975) shows that, provided that u 2 C2, a necessary and sufficient condition for multivariate risk aversion is @ 2u/@xi@xj 6 0 for i – j. Multivariate risk aversion is also characterized by Richard (1975) in terms of the signs of the partial derivatives. The alternating sign condition of Richard (1975) implies the submodularity of the corresponding utility function (see Ortega and Escudero, 2010). Tsetlin and Winkler (2009) show that a decision-maker ‘‘to be consistent with preferring to combine good with bad should have a utility function with alternating signs for successive partial

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derivatives.” Theorem 1 in Tsetlin and Winkler (2009, p. 1945) states that a decision-maker prefers to combine good with bad if and only if its utility function belongs to UBb :UBb is the class of all ‘‘B-dimensional real-valued functions for which all partial derivatives of a given order up to order b have the same sign, and that sign alternates, being positive for odd orders and negative for even orders. Tsetlin and Winkler (2009, p. 1945),” UBb plays an important role also in the definition of a new class of utility functions in Prekopa and Madi-Nagy (2008). In respect of Richard (1975) and Tsetlin and Winkler (2009) define the preference for combining good lotteries with bad lotteries as a ‘‘stronger condition, encompassing single-attribute risk aversion and going beyond it.” This condition is expressed by the additional assumption that the decision-maker is single-attribute risk averse, e.g.,

@ 2 u=@x2i @u=@xi

P 0. @ 2 u=@x2

Now, we show that the assumptions u 2 UBb and @u=@xi i P 0 stated in Tsetlin and Winkler (2009) imply the neg-ultramodularity of u. In Tsetlin and Winkler (2009) (but also in Richard (1975)), u is increasing in the attributes, whence @u/@xi P 0, "i. u 2 UBb then implies that the second order partial derivatives are negative, i.e., @ 2u/ @xi@xj 6 0, "i, j. Note that i does not need to be different from j due to the single-attribute risk aversion assumption. Now, by Theorem 5.5 in Marinacci and Montrucchio (2005), a function u 2 C2 is negultramodular iff @ 2u/@xi@xj 6 0, "i, j. Therefore:

tation (HDMR) theory. Rabitz and Alis (1999) prove the functional ANOVA decomposition through the dissection of the linear space to which the function belong. A fourth and independent way of proving the functional ANOVA expansion is due to Sobol (1969) and Sobol (1993). In Sobol (1969), the decomposition is developed in the context of quadrature methods and called ‘‘the decomposition into summands of different dimensions (Owen, 2003, p. 2). In Sobol (1993), the uniqueness of Eq. (20) is proven via nested integrations.1 In this work, we utilize the framework of Rabitz and Alis (1999). Without loss of generality, one refers to a scalar function f : In # Rn ! R, where I denotes the unit interval [0, 1], and In the n-dimensional unitary hypercube. (In, BðIn Þ, l) is a Borel measure space. For the sake of notation simplicity in the remainder, l denotes the Lebesgue measure (on the standard choices of In and the Lebesgue measure, see also Rabitz and Alis, 1999; Sobol, 2001; Owen, 2003, Sobol, 2003; Wang, 2006; Malliavin, 1995, p. 224–227). f belongs to a linear vector space of functions denoted by F. We require f to be at least measurable.2 Rabitz and Alis (1999) then utilize the following decomposition of F (a similar approach is also used in Takemura (1983)). Lemma 1 ( Rabitz and Alis, 1999).

F ¼ F0 

3. High dimensional model representation theory and functional ANOVA In this section, we describe the foundations of functional ANOVA and offer its interpretation as a tool for expanding multivariate measurable functions. The origin of functional ANOVA is linked to the problem of decomposing the variance of a square integrable statistics generated by the seminal works of Hoeffding in the 1940’s (Hoeffding, 1948). The ‘‘jackknife” decomposition is proven in Efron and Stein (1981). The technique utilized by Efron and Stein (1981) consists in the dissection of f via a sequence of nested conditional expectations in accord with a Grahm–Schmidt orthogonalization process. Orthogonality is at the basis of the results by Takemura (1983), where the orthogonal decomposition of any square integrable function is cast in the context of tensor analysis and multilinear algebra. Rabitz and Alis (1999) introduce an alternative generalization of functional ANOVA, called high dimensional model represen-

Fi 

i¼1

2

Proposition 2. Let u 2 C (X1  X2      Xn). If and only if the decision-maker is averse to any multivariate risk in the sense of Tsetlin and Winkler (2009), then u is neg-ultramodular. One notes that, by definition of ultramodularity, the above proposition holds also if u is monotonically decreasing, with negultramodularity replaced by ultramodularity. As far as elicitation is concerned, we refer to Tsetlin and Winkler (2009) (in particular, Section 5, p. 1948), where the assessment of a utility function displaying the above-described multivariate risk attitudes is discussed. One notes that both our work up to this point, and the preceding literature, has characterized monotonicity and ultramodularity in terms of first and second order derivatives. For this characterization to hold, u must be twice differentiable. Quantitative elicitation methods (see, for instance the GRIP method in Greco et al. (2008) and Figueira et al. (2009)), however, lead to utility functions that are, generally, non-differentiable. Hence, one needs to obtain a more general characterization of the properties of u, not relying on differentiation. In the next section, the use of functional ANOVA as a way for expanding a multivariate function without resorting to regularity assumptions is discussed.

n X

X

F i;j      F 123...n ;

ð3Þ

i
where  is the direct-sum operator and the subspaces are defined as

8 F 0  ff 2 F : f ¼ a; a 2 Rg > > R > < F i  ff 2 F : f ¼ fi ðxi Þ with I fi ðxi Þdli ¼ 0g R > > F i;j  ff 2 F : f ¼ fi;j ðxi ; xj Þ with I fi;j ðxi ; xj Þ dls ¼ 0; s ¼ i; jg > : ... ð4Þ The following theorem then follows by projection (see Rabitz and Alis, 1999). Theorem 1 (Sobol, 1993; Rabitz and Alis, 1999). Let f : In # Rn ! R, f 2 F. Under the above assumptions, the following decomposition of f in Eq. (5) is unique:

f ðxÞ ¼ f0 þ

n X

X

fi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ

ð5Þ

k¼1 i1
with

f0 ¼ M0 f ¼ El ½f ðxÞ fi ðxi Þ ¼ Mi f ðxÞ  f0 fi1 ;i2 ðxi1 ; xi2 Þ ¼ Mi1 ;i2 f ðxÞ  fi1 ðxi1 Þ  fi2 ðxi2 Þ  f0 ...

ð6Þ

and

  Mi1 ;i2 ;...;ik ½f  ¼ El f jxi1 ; xi2 ; . . . ; xik :

ð7Þ

In Eq. (5), Mi1 ;i2 ;...;ik are the conditional expectations of f given xi1 ; xi2 ; . . . ; xik , and are referred to in Rabitz and Alis (1999), Alis and Rabitz (2001) and Sobol (2003) as projection operators. In the remainder, the notation

  mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ ¼ El f jxi1 ; xi2 ; . . . ; xik ;

ð8Þ

1 In the remainder, the terms functional ANOVA, HDMR and integral decomposition shall be regarded as synonyms. 2 Given the probability space ðX; BðXÞ; lÞ; Lp ðX; BðXÞ; lÞ denotes the set of all R l  p measurable functions w : X ! Rn such that X kwðxÞkp dlðxÞ < 1 (Malliavin, 1995).

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shall display the dependence of Mi1 ;i2 ;...;ik ½f  on xi1 ; xi2 ; . . . ; xik . We call the functions mi1 ;i2 ;...;ik in Eq. (8) non-orthogonalized terms. The functions fi are referred to as first order terms (Rabitz and Alis, 1999; Alis and Rabitz, 2001; Sobol, 2003) or main effects (Huang, 1998; Wang, 2006; Hooker, 2007), the functions fi1 ;i2 ; . . . ;ik as interaction terms or interaction effects. One needs to recall the crucial role played by the independence assumption in granting uniqueness to the functional ANOVA expansion. In fact, it is proven by Bedford (1998) that, if correlations are present, the decomposition depends on the initial lexicographical ordering of the random variables. The same f then shares multiple representations and Eq. (5) looses its uniqueness. Without claiming exhaustiveness, one can group the numerous applications of functional ANOVA into three interrelated fields: high-dimensional integration, high-dimensional model representation and global sensitivity analysis. In Wang (2006), functional ANOVA is utilized as a dimension-reduction technique for integration problems in financial applications. Functional ANOVA in high-dimensional integration has been studied also in Sobol (1969), Sobol (1993), Sobol (2001), Sobol (2003), Owen (2003) and Sobol et al. (2007). In Rabitz and Alis (1999), Alis and Rabitz (2001) and Li et al. (2001), functional ANOVA is employed ‘‘for improving the efficiency of deducing high dimensional input–output system behavior (Alis and Rabitz, 2001, p. 1).” In these works, the functional ANOVA expansion is utilized as an interpolation method, in a meta-modelling scheme, and originates the so-called HDMR theory (Rabitz and Alis, 1999). In particular, Rabitz and Alis (1999) and Alis and Rabitz (2001) introduce the cut-HDMR expansion, which provides as one of the most useful tools for complex model output interpolation and dimension reduction. A third and interrelated field where functional ANOVA has been widely applied is global sensitivity analysis of model output (Wagner, 1995; Homma and Saltelli, 1996; Saltelli et al., 2000; Saltelli et al., 1999; Saltelli and Tarantola, 2002; Borgonovo, 2006; Borgonovo, 2010). Indeed, the first application of functional ANOVA in OR is in global sensitivity analysis, by Wagner (1995). Notable efforts have since then been performed towards refining the numerical estimation of the terms in the functional ANOVA expansion, especially in connection with the of Sobol’ global Sensitivity indices (see Homma and Saltelli, 1996; Saltelli et al., 1999 introduce the use of Fourier Amplitude Sensitivity Test to estimation main effects). A common feature across all the above mentioned work is the assumption of independence among the random variables. Bedford(1998) shows that this assumption is central to obtain the uniqueness of the functional ANOVA representation. The work by Hooker (2007) provides a generalization of functional ANOVA for the diagnostics of high dimensional models in the presence of dependent variables. In this work, we utilize functional ANOVA as a function expansion method, namely emphasis is posed on Eq. (5). In many works related to functional ANOVA, f is required to be square integrable, i.e., f 2 L2 ðX; BðXÞ; lÞ, since the variance of f is the goal of the analysis. In our work, because we are concerned with Eq. (5), the assumption can be softened to f 2 LðX; BðXÞ; lÞ. As we are to see, this allows us to obtain links between properties of u(x) and those of its conditional utilities, bypassing requirements on the regularity of u(x) implied by a methodology based on first or second order partial derivatives. Such need stems from the fact that, generally, an elicited utility function is not differentiable (see the GRIP method in Greco et al. (2008) and Figueira et al. (2009)). However, in spite of the widespread utilization of functional ANOVA, properties as ultramodularity and monotonicity have not been studied via this type of integral expansion yet. It is then the purpose of the next sections to investigate how-whether these properties are maintained in functional ANOVA. We start with monotonicity.

4. Monotonicity properties in functional ANOVA The definition of non-decreasing function of interest for our work is as follows. Definition 2. f : In ! R is non-decreasing on In, if, "x and "y 2 In such that x 6 y, then

f ðxÞ 6 f ðyÞ:

ð9Þ

If the strict inequality holds, one says that f is increasing. Similarly, f is non-increasing if "x and "y 2 In such that x 6 y

f ðxÞ P f ðyÞ:

ð10Þ

If the strict inequality holds, one says that f is decreasing. Let us assume that Eqs. (5) and (6) hold, e.g., f is at least measurable. The following result holds (see Appendix A for the proof). Lemma 2. If f is non-decreasing, then all non-orthogonalized terms mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ in its ANOVA expansion are. By recalling the definition of mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ, Lemma 2 states that conditional expectation preserves the monotonicity of f. The following result concerns the monotonicity of the main effects of the functional ANOVA expansion (see Appendix A for the proof). Theorem 2. If f is non-decreasing, then all first order terms in its ANOVA expansion are non-decreasing. Given Definition 2, it is readily seen that results similar to Lemma 2 and Theorem 2 hold in the case of f increasing, nondecreasing and strictly decreasing. Hence, monotonicity is preserved by the main effects (first order terms) of the functional ANOVA expansion. For higher order projections, this property does not hold, in general. The terms fi1 ;i2 ;...;ik are the result of an orthogonalization process involving the difference between f and lower order terms. As an example, second order terms are given by:

fr;l ðxr ; xl Þ ¼ mr;l ðxr ; xl Þ  fr ðxr Þ  fl ðxl Þ  f0 :

ð11Þ

In spite of the fact that mr,l (xr, xl), fr(xr), fl(xl) are all non-decreasing individually—if f is—the subtraction of fr(xr) and fl(xl) does not insure that

fr;l ðxr ; xl Þ < fr;l ðxr þ hr ; xl þ hl Þ;

8h r ; h l :

ð12Þ

However, the inequality in (12) is satisfied if3:

Dfr þ Dfl 6 Dmr;l :

ð15Þ

In Eq. (15) states that f r,l(x r, x l) is non-decreasing when the change in f given the simultaneous changes in xr and xl averaged over all possible values of the remaining variables (Dmr,l) is greater than the sum of the average changes due to the variations in xr and xl individually (Dfr + Dfl). The above result can be extended to higher-order terms as follows (see Appendix A for the proof). Theorem 3 (Sufficient condition for monotonicity). If f is nondecreasing, then the generic kth order term fxi1 ;xi2 ;...;xi ðxi1 ; xi2 ; . . . ; xik Þ k (1 < k 6 n) in its ANOVA expansion is non-decreasing if the following condition holds:

3

Starting with the inequality

mr;l ðxr ; xl Þ  fr ðxr Þ  fl ðxl Þ  f0 6 mr;l ðxr þ hr ; xl þ hl Þ  fr ðxr þ hr Þ  fl ðxl þ hl Þ  f0 ð13Þ one obtains:

fr ðxr þ hr Þ  fr ðxr Þ þ fl ðxl þ hl Þ  fl ðxl Þ 6 mr;l ðxr þ hr ; xl þ hl Þ  mr;l ðxr ; xl Þ

ð14Þ

330

Dmi1 ;i2 ;...;ik P

F. Beccacece, E. Borgonovo / European Journal of Operational Research 210 (2011) 326–335 k1 X

X

Dfi1 ;i2 ;...;is

ð16Þ

s¼1 i1
where

  Dmi1 ;i2 ;...;ik ¼ mi1 ;i2 ;...;ik xi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik    mi1 ;i2 ;...;ik xi1 ; xi2 ; . . . ; xik Dfi1 ;i2 ;...;ik ¼ fi1 ;i2 ;...;ik ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik Þ    fi1 ;i2 ;...;ik xi1 ; xi2 ; . . . ; xik

Theorem 5 (Sufficient condition for ultramodularity of higher order terms). If f is ultramodular, then the generic k th order term fxi1 ;xi2 ;...;xi ðxi1 ; xi2 ; . . . ; xik Þð1 < k 6 n) in the decomposition is ultrak modular provided that the following condition holds:

Dmi1 ;i2 ;...;ik ðyÞ  Dmi1 ;i2 ;...;ik ðxÞ ð17Þ

and h P 0. Theorem 3 states the conditions on f such that higher order terms in the functional ANOVA expansion retain the monotonicity of f. In the elicitation of multiattribute utility functions, however, it is often assumed (required) a separable form of the utility function. In particular, additive independence results in an additive form of the utility function, while mutual utility independence results in a multiplicative form (Keeney and Raiffa, 1993). We recall that separable multiattribute utility functions play also a relevant role in simplifying the algorithms for solution of multiattribute decisionanalysis problems represented in the form of influence diagrams, as discussed in Tatman and Shachter (1990). Let us then investigate monotonicity properties of additive and multiplicative functions via functional ANOVA. We have the following results (the proofs are in Appendix A.)   P Corollary 1. Let f : In # Rn ! R be additive f ðxÞ ¼ ni¼1 zi ðxi Þ . It is then true that f is non-decreasing if and only if zi(x i) are nondecreasing, "i. Q Corollary 2. Let f : In # Rn ! Rbe of the form f ðxÞ ¼ zi ðxi Þ with z n i P 0 "i. Assume that f is continuous on I . Then, f is non-decreasing, if and only if z i(x i)" i are non-decreasing. Corollaries 1 and 2 set forth the conditions under which the monotonicity of the univariate functions zi(xi) insure the monotonicity of f, when f is additive or multiplicative. In the next section, we are to address ultramodularity properties. 5. Ultramodularity in Functional ANOVA This section is devoted to the functional ANOVA expansion of ultramodular functions. In particular, we investigate what are the conditions under which ultramodularity is preserved at the various orders of the expansion. The next result characterizes first order terms. Theorem 4. If f is ultramodular, then the first order terms of Eq. (5), fi, i = 1, 2, . . . , n, are ultramodular. Theorem 4 states that the ultramodularity of f is a sufficient condition for the ultramodularity of the first order terms of the integral decomposition. To insure the ultramodularity of higher order terms (e.g., fi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ), however, a further assumption on the ‘‘strength” of the ultramodularity of f must be added. We start with proving the following result for the non-normalized terms in Eqs. (5) and (6).

P

k1 X

X

s¼1 i1
Dfi1 ;i2 ;...;is ðyÞ 

k1 X

X

Dfi1 ;i2 ;...;is ðxÞ

ð18Þ

s¼1 i1
where

  Dmi1 ;i2 ;...;ik ðyÞ ¼ mi1 ;i2 ;...;ik yi1 þ hi1 ; yi2 þ hi2 ; . . . ; yik þ hik    mi1 ;i2 ;...;ik yi1 ; yi2 ; . . . ; yik ;   Dmi1 ;i2 ;...;ik ðxÞ ¼ mi1 ;i2 ;...;ik xi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik  mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ;   Dfi1 ;i2 ;...;is ðyÞ ¼ fi1 ;i2 ;...;is yi1 þ hi1 ; yi2 þ hi2 ; . . . ; yis þ his

ð19Þ

 fi1 ;i2 ;...;ik ðyi1 ; yi2 ; . . . ; yis Þ;   Dfi1 ;i2 ;...;is ðxÞ ¼ fi1 ;i2 ;...;is xi1 þ hi1 ; xi2 þ hi2 ; . . . ; xis þ his    fi1 ;i2 ;...;ik xi1 ; xi2 ; . . . ; xis : Theorem 5 states the conditions under which all terms in the integral expansion of a ultramodular function are ultramodular. In this respect, it parallels Theorem 3. The conditions implied by Theorem 5 can be softened in the case the utility function is separable. Concerning the ultramodularity of additive functions, we prove in Appendix A the following results.   P Corollary 3. Let f : In # Rn ! R be additive f ðxÞ ¼ ni¼1 zi ðxi Þ f is ultramodular if and only if z i(x i) are ultramodular "i. Corollary 3 allows us to extend a result in Proposition 4.4. In Marinacci and Montrucchio (2005), where a sufficient condition for an additive function f to be ultramodular and convex is given by the convexity of each zi "i. P Corollary 4. Let f (x) be of the form ni¼1 zi ðxi Þ, f(x) is ultramodular if and only if it is convex. The next result concerns the ultramodularity of multiplicative functions. Q Corollary 5. If f ¼ ni¼1 zi ðxi Þ is ultramodular and zi(xi) > 0, then zi(xi) is ultramodular "i = 1, 2, . . . , n. If, in addition, zi(xi) is increasing "i = 1, 2, . . . , n, the converse is also true. Corollary 5 states that, if f(x) is multiplicative, then its ultramodularity insures the ultramodularity of its univariate functions. However, the ultramodularity of the univariate functions does not insure the ultramodularity of f, unless the univariate functions are also increasing. The findings of the present section and of Section 4 concern a generic multivariate f. In the next two sections, we discuss the MAUT implications of these findings. 6. Functional ANOVA of multiattribute utility functions

Lemma 3. If f is ultramodular, then all non-orthogonalized terms mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ (2 6 k 6 n) in its ANOVA expansion are ultramodular.

In this section, we discuss the multiattribute utility theory implications of the findings of Sections 3–5. Theorem 1 has the following implication.

The sufficient condition under which all the orthogonalized terms in Eq. (5), (e.g., the functions fi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ; 2 6 k 6 n) are ultramodular is given below.

Corollary 6. Any measurable multiattribute uðxÞ : X 1  X 2     X n ! R, can be written as:

utility

function

F. Beccacece, E. Borgonovo / European Journal of Operational Research 210 (2011) 326–335

uðxÞ ¼ u0 þ

n X

ui ðxi Þ þ

X

ui;j ðxi ; xj Þ þ    þ u1;2;...n ðx1 ; x2 ; . . . ; xn Þ:

i
i¼1

ð20Þ Corollary 6 (Eq. (20)) states that any multiattribute utility function can be expanded in a series of terms in which first order terms concern preferences over individual attributes, the second order terms concern interactions among pairs of attributes, the third order terms interactions among triplets, etc. We observe that Eq. (20) applies without reference to any particular preference structure. Let us investigate this aspect further. We start by an observation of Rabitz and Alis (1999, p. 198). The observation concerns complex mathematical models and addresses the ‘‘curse of dimensionality,” namely the presence of high-order interactions: ‘‘a dramatic reduction in this scaling (the number of interactions) is often expected to arise (. . .) due to the presence of only low-order correlations amongst the input variables having a significant impact upon the output.” In applications of functional ANOVA to complex numerical models (Wang, 2006; Rabitz and Alis, 1999; Li et al., 2001), one can cut Eq. (20) at a predetermined order to reduce model complexity. For instance, in a second order cut, one assumes

uðxÞ ’ u0 þ

n X i¼1

ui ðxi Þ þ

X

ui;j ðxi ; xj Þ:

ð21Þ

i
When financial, physical or chemical systems are concerned, the only way to ascertain the accuracy of the approximation in Eq. (21) is by performing numerical experiments. In MAUT, however, one can utilize assumptions on the preference structure of the decision-maker to determine a priori the order of the expansion. This marks a notable departure between the traditional utilization of functional ANOVA and its utilization in MAUT. Furthermore, as we are to see, the presence or absence of interactions in u(x) is linked to the multivariate risk attitudes of the decision-maker. In many applications of multiattribute decision-making, it is desirable to quantify u via the elicitation of the conditional utilities in each attribute (Keeney and Raiffa, 1993; Keeney, 2006; Baucells et al., 2008; Greco  et al.,  2008; Figueira et al., 2009). In other words, one assesses xi jx0ðiÞ for different levels of xi, with the remaining  u  attributes x0ðiÞ fixed at a given level x0ðiÞ . In these applications, a separable form for u is often assumed (Greco et al., 2008; Figueira et al., 2009; Baucells et al., 2008). In the remainder of the work, we address the questions of whether, given that u is monotonic/ultramodular, all the conditional utilities need to be monotonic/ultramodular and whether the converse statement needs to be true. A positive answer, would insure that, if all the elicited conditional utilities are monotonic/ultramodular, then the decision-maker is characterized by a monotonic/ultramodular utility functions. The converse statement would imply the following. If one assume that a decision-maker is characterized by a monotonic/ultramodular utility function, then none of the elicited conditional utilities can be non-monotonic/non-ultramodular, without contradicting the hypothesis on the decision-maker’s preferences. In the next sections, we show that, using functional ANOVA to decompose u, it is possible to find exact relations between the form of u(x) and the conditional utilities, without relying on the regularity of u(x). 6.1. Additive independence, ultramodularity and monotonicity Additive independence is invoked by many works in multiattribute decision-making (see Keeney and Raiffa, 1993; Zopounidis and Doumpos, 2002; Baucells et al., 2008). The notion of additive independence originates in Fishburn (1965) and refers to the case in which there is no interaction of

331

preference among a set of attributes. For the sake of notation simplicity, let us refer to the two-attribute case. X and Y are additive independent if the comparison of two arbitrary lotteries defined by two joint probability distributions on X  Y depends only on their marginal probability distributions (Keeney and Raiffa, 1993). As proven in Fishburn (1965), X and Y are additive independent if and only if the utility function u(x, y) is additive, i.e., if and only if

uðx; yÞ ¼ uðx0 ; yÞ þ uðx; y0 Þ;

ð22Þ

where u(x0, y) and u(x, y0) are called conditional utility functions. All the utility functions are normalized. The findings in Sections 4 and 5, in particular Corollaries 1 and 3, allow us to derive the following results concerning the monotonicity and ultramodularity of u(x, y) [the proof is in Appendix A]. Theorem 6. Consider a utility function under additive independence. Then: 1. u (x, y) is non-decreasing if and only if u (x0, y) and u (x, y 0) are. 2. u (x, y) is neg-ultramodular if and only if u (x0, y) and u (x, y0) are. 3. if u (x, y) is neg-ultramodular, then it is concave. Point 1 in Theorem 6 implies that, if in an elicitation problem, additive independence is assumed, then non-satiation in each attribute implies multivariate non-satiation of the decision-maker. Points 2 implies that a decision-maker who is single-attribute risk averse, is also multivariate risk averse in the sense of Tsetlin and Winkler (2009). In fact, if and only if the conditional utility functions are concave (one recalls that for univariate functions neg-ultramodularity and concavity coincide), then the multiattribute utility function is neg-ultramodular. Point 3 states that if a decision maker is multivariate risk averse, then, under additive independence, She is also globally risk averse (in fact, her multiattribute utility function, when neg-ultramodular is also concave). We recall that, in general, single-attribute risk aversion or multivariate risk-aversion are not sufficient for global risk aversion. Keeney and Raiffa (1993, p. 253) suggest additive independence as way for approximating utility functions. This idea can be read through Eq. (20) as follows: additive independence is equivalent to a cut at order 1 of the functional ANOVA expansion of a generic u:

uðxÞ ’

n X

ui ðxi Þ

ð23Þ

i¼1

Under this cut, no interactions are present in the utility function. We recall that by Theorem 9 in Richard (1975, p. 20) the decision-maker is multivariate risk neutral if and only if P uðxÞ ¼ ni¼1 ui ðxi Þ. Hence, multivariate risk-neutrality is associated with the absence of interactions among attributes. In the next section, we explore the case in which interactions among attributes are present, namely the case of utility independence. 6.2. Mutual utility independence, ultramodularity and multi-attribute risk aversion In multiattribute theory, the assumption of additive independence may result too restrictive (Hogart and Karelaia, 2005). A first relaxation of such assumption is represented by utility independence. Utility independence has attracted a widespread attention in literature for several reasons. On the one hand, the utility independence assumptions are appropriate in many realistic problems (Keeney, 2006). On the other hand, utility independence helps in structuring the problem and in performing sensitivity analysis.

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Utility independence allows the multiplicative representation of the utility function u(x,y). If X and Y are mutually utility independent, then u(x,y) can be expressed by the multilinear representation (Fishburn, 1980; see also Keeney and Raiffa, 1993, Theorem 5.2, p. 234)

uðx; yÞ ¼ uðx; y0 Þ þ uðx0 ; yÞ þ kuðx; y0 Þuðx0 ; yÞ

ð24Þ

where u, u(x, y0), u(x0,y) are based on the same common origin and consistently scaled by the constant k 2 R. By algebraic manipulation, one obtains (Keeney and Raiffa, 1993, p. 238, Eq. 5.31)

kuðx; yÞ þ 1 ¼ ½kuðx; y0 Þ þ 1½kuðx0 ; yÞ þ 1

ð25Þ

The above procedure is extended to the n  attribute case (Keeney and Raiffa, 1993, Eq. (6.21), p. 291), and leads to

kuðxÞ þ 1 ¼

n Y

½kuðxi ; ^x0 Þ þ 1

  where ðxi ; ^ x0 Þ ¼ x01 ; x02 ; . . . ; xi ; . . . ; x0n . We also recall the equivalent multiplicative representation

uðzÞ ¼ a þ b

ui ðzi Þ

Theorem 8. Let X = X 1  X 2 . . .X n, be a set of attributes over which the decision-maker expresses preferential independence. Then, let

v ðxÞ ¼

n X

v i ðxi Þ

the value function. (The additive form follows by preferential independence.) Let u (x) the corresponding utility function. Let X i be utility independent. One has 1. If 2. If

vi(x i) is non-decreasing "i, then u (x) is non-decreasing. v i(x i) is ultramodular "i, then u (x) is ultramodular.

ð27Þ

We recall that, by Theorem 6.11 (Keeney and Raiffa, 1993, p. 330), the utility function corresponding to the preference function in Eq. (29) is either of the multiplicative type

uðxÞ ecv ðxÞ ¼

(b > 0) studied in Richard (1975). In summary, under utility independence the multiattribute utility function is multiplicative. Its monotonicity and ultramodularity properties can, then, be derived by exploiting the results of Sections 4 and 5 on the monotonicity and ultramodularity properties of generic multiplicative functions [Corollaries 2 and 5 are utilized in the Appendices to prove the next results.]. The following result holds. Theorem 7. Let u be increasing in the attributes. Then, u (x) is ultramodular (neg-ultramodular) if and only if u i(x i) is ultramodular (neg-ultramodular). We recall that, by Proposition 2, a decision-maker associated with a multiattribute ultramodular function displays the multivariate risk attitudes of Tsetlin and Winkler (2009). By comparing Eqs. (22) and (24), one notes that interactions are present in Eq. (24). For simplicity, consider a three attribute case. In this case, the functional ANOVA expansion of u(x) writes: 3 X

ui ðxi Þ þ u1;2 ðx1 ; x2 Þ þ u1;3 ðx1 ; x3 Þ þ u2;3 ðx2 ; x3 Þ þ u123 ðx1 ; x2 ; x3 Þ

i¼1

ð28Þ The terms u1,2 (x1, x2) + u1,3(x1, x3) + u2,3(x2, x3) + u123(x1, x2, x3) are null if u is additive, and the decision-maker is multivariate riskneutral. Conversely, they all appear as soon as interactions among attributes are present. In this latter case, the decision-maker is no more multivariate risk neutral. In fact, given any twicedifferentiable function f, then a non-additive form of f implies non-null mixed partial derivatives. In the next section, we present results for a second relaxation of the additive utility assumption, namely, preferential independence. 6.3. Preferential independence, ultramodularity and monotonicity A further relaxation of the additive independence assumption is represented by preferential independence. Keeney and Raiffa (1993) provide the procedure for obtain a multiattribute utility function given that an additive value function has been assessed. The corresponding utility function can be of an additive or multiplicative form. The next result sets forth the conditions under which, in preferential independence framework, ultramodularity

n Y

ecv i ðxÞ

ð30Þ

i¼1

i¼1

u¼

ð29Þ

i¼1

ð26Þ

i¼1

n Y

and monotonicity of the marginal value functions insure ultramodularity and monotonicity of the multiattribute utility function. The proof is in Appendix A.

or proportional to v(x), i.e.,

uðxÞ v ðxÞ:

ð31Þ

Theorem 8 then states that monotonicity and ultramodularity of the assessed value function are transferred into the utility function, in both the multiplicative and additive representations. Therefore, the multivariate risk attitudes of the decision-maker are characterized directly by the value function. 7. Conclusions In this work, we have presented a formal approach for connecting the properties of a multiattribute utility function to those of its conditional utilities, without relying on regularity assumptions. In the quantitative assessment of multiattribute utility functions, procedures involving elicitation by aggregation of conditional utilities play a central role. The elicited univariate utility functions are not necessarily regular. As a consequence, the characterization of non-satiation (monotonicity) and risk aversion (concavity, ultramodularity) properties of u by first and second order partial derivatives is not possible. We have seen that the high dimensional model representation theory, by requiring the sole measurability of u, provides the required generalization of the mathematical framework. However, in spite of the widespread utilization of functional ANOVA, its monotonicity and ultramodularity properties have not been formalized yet. It has then been necessary to investigate monotonicity and ultramodularity properties for generic multivariate functions (f) first. As far as non-orthogonalized terms in the integral expansion of f are concerned, findings show that they possess the same monotonicity (ultramodularity) properties of f. As far as orthogonalized terms are concerned, main effects (fi) always possess the same monotonicity properties of f. However, this is not the case for higher order terms due to the orthogonalization procedure. We have then derived the sufficient conditions on f that insure that all terms in the ANOVA expansion possess the same monotonicity (ultramodularity) properties of f. The differences between the traditional utilization of functional ANOVA and its utilization in MAUT have been investigated next. In particular, in MAUT, it is possible to determine a priori the order of the expansion by the assumptions on the decision-maker preferences. The link between the presence of interactions in the functional ANOVA expansion of u and the multivariate risk attitudes of the

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decision-maker has also been discussed. We have seen that multivariate risk neutrality is equivalent to a cut of order 1 in the expansion, i.e., it implies the absence of interactions among attributes. When some degree of interactions is allowed, we have shown that multivariate risk attitudes of Tsetlin and Winkler (2009)—preferences for combining good with bad—imply the neg-ultramodularity of the utility function. The findings have then been specialized to the cases of separable (additive and multiplicative) utility functions. We have seen that under utility independence, u is monotonic (ultramodular) if and only if all the corresponding single-attribute conditional utility functions are. Under mutual utility independence, if u is increasing, then it is ultramodular (monotonic) if and only if all the conditional utility functions are. Under preferential independence, letting v denote the multiattribute value function for preferences under certainty, we have seen that if v is non-decreasing, then u is; if v is ultramodular, then u is. The above results hold without restrictions on the regularity of u.

Proof of Theorem 3. By Eq. (6), we have:

fi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ ¼ mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ 

k1 X

X

fi1 ;i2 ;...;is ðxi1 ; xi2 ; . . . ; xis Þ  f0 ;

s¼1 i1
ð38Þ fi1 ;i2 ;...;ik ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik Þ ¼ mi1 ;i2 ;...;ik ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik Þ 

k1 X

X

fi1 ;i2 ;...;is ðxi1 þ hi1 ; . . . ; xis þ his Þ  f0 :

ð39Þ

s¼1 i1
Thus:

fi1 ;i2 ;...;ik ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik Þ P fi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ ð40Þ is equivalent to

  mi1 ;i2 ;...;ik xi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik

Acknowledgments The authors thank the anonymous referees for very perceptive and insightful comments that have greatly helped us in improving the manuscript. We also thank the participants of the XIII International Conference on the Foundations and Applications of Utility, Risk and Decision Theory, IESE, Barcelona, July 2008, and of the AMASES 2007 conference for their insightful comments and observations.



k1 X

X

fi1 ;i2 ;...;is ðxi1 þ hi1 ; . . . ; xis þ his Þ  f0

s¼1 i1
  P mi1 ;i2 ;...;ik xi1 ; xi2 ; . . . ; xik 

k1 X

X

fi1 ;i2 ;...;is ðxi1 ; xi2 ; . . . ; xis Þ  f0 ;

ð41Þ

s¼1 i1
which, rearranged leads to:

mi1 ;i2 ;...;ik ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik Þ  mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ Appendix A. Proofs

P

I

nk

f ðxÞ

Y

  fi1 ;i2 ;...;is ðxi1 ; xi2 ; . . . ; xis Þ :

ð42Þ



ð32Þ

In particular, when h ¼ ½0 . . . hi1 . . . hik . . . 0, with his P 0; s ¼ 1; 2; . . . ; k (k < n), Eq. (32) holds. Let us then integrate both sides of Eq. (32) with respect to all xi’s but xi1 ; xi2 ; . . . ; xik . By the monotonicity property of the integration operation, we get:

Z

X  fi1 ;i2 ;...;is ðxi1 þ hi1 ; . . . ; xis þ his Þ

s¼1 i1
Proof of Lemma 2. Let y = x + h. Then, "h P 0 (i.e., hi P 0, i = 1, 2,. . ., n), by (9), we have:

f ðxÞ 6 f ðx þ hÞ:

k1 X

dxi 6

Z I

i–i1 ;i2 ;...;ik

nk

f ðx þ hÞ

Y

dxi ;

ð33Þ

i–i1 ;i2 ;...;ik

which implies:

mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ 6 mi1 ;i2 ;...;ik ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik Þ:

 ð34Þ

Proof of Theorem 4. Let y = x + h. Then, Eq. (1) holds for any h, and, therefore, in particular when h = [0 . . . hi . . . 0], with hi P 0. Thus, it holds that

f ðxi þ hi ; ^xÞ  f ðxi ; ^xÞ 6 f ðyi þ hi ; ^xÞ  f ðyi ; ^xÞ;

ð43Þ

where ^x ¼ x=xi ¼ ðx1 ; x2 ; . . . ; xi1 ; xiþ1 ; . . . ; xn Þ; hi > 0; xi 6 yi . Let us then integrate both sides of (43) on all xk, but xk = xi. Since the component of the Sobol’ decomposition are given by mi ðxi Þ ¼ f0 þ R Q fi ðxi Þ ¼ In1 f ðxÞ k–i dxk , it holds

mi ðxi þ hi Þ  mi ðxi Þ 6 mi ðyi þ hi Þ  mi ðyi Þ: Since fi(xi) = mi(xi)  f0, by point (b) of Proposition 1, we get

fi ðxi þ hi Þ  fi ðxi Þ 6 fi ðyi þ hi Þ  fi ðyi Þ:

ð44Þ



Proof of Theorem 2. Let y = x + h. Then, by (9), we have:

f ðxÞ 6 f ðx þ hÞ

ð35Þ

"h P 0 (i.e., hi P 0, i = 1, 2, . . ., n). Eq. (32) holds, in particular, when h = [0 . . . hi . . . 0], with hi P 0. Let us then integrate both sides of Eq. (32) with respect to all xk’s but xi. We get:

Z I

n1

f ðxÞ

Y

dxk 6

k–i

Z I

n1

f ðx þ hÞ

Y

dxk ;

ð36Þ

k–i

which proves the assertion. h

f ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik ; ^xÞ  f ðxÞ ^Þ  f ðyÞ; 6 f ðyi1 þ hi1 ; yi2 þ hi2 ; . . . ; yik þ hik ; y

ð37Þ

ð45Þ

where ^x ¼ x=xi1 ; xi2 ; . . . ; xik in this case. Integrating both sides of Eq. (45) with respect to all xi’s but xi1 ; xi2 ; . . . ; xik , thanks to the monotonicity property of the integration operation, we get:

Z

which implies:

mi ðxi Þ  f0 6 mi ðxi þ hÞ  f0 ;

Proof of Lemma 3. Let y = x + h, with h ¼ ½0 . . . hi1 . . . hi2 . . . hik . . ., where his > 0, s = 1, . . . , k, and 2 < k < n. Then, by Eq. (1), we have:

Ink

½f ðx þ hÞ  f ðxÞ

Y i–i1 ;i2 ;...;ik

dxi 6

Z Ink

½f ðy þ hÞ  f ðyÞ

Y

dyi ;

i–i1 ;i2 ;...;ik

ð46Þ

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F. Beccacece, E. Borgonovo / European Journal of Operational Research 210 (2011) 326–335

which implies:

mi ðxi Þ ¼ zi ðxi Þ 

mi1 ;i2 ;...;ik ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik Þ  mi1 ;i2 ;...;ik ðxi1 ; xi2 ; . . . ; xik Þ 6 mi1 ;i2 ;...;ik ðyi1 þ hi1 ; yi2 þ hi2 ; . . . ; yik þ hik Þ  mi1 ;i2 ;...;ik ðyi1 ; yi2 ; . . . ; yik Þ:

ð47Þ



Proof of Theorem 5. Let us start with Eq. (47). In order for fi1 ;i2 ;...;ik to be ultramodular, it must hold that:

  fi1 ;i2 ;...;ik ðxi1 þ hi1 ; xi2 þ hi2 ; . . . ; xik þ hik Þ  fi1 ;i2 ;...;ik xi1 ; xi2 ; . . . ; xik   6 fi1 ;i2 ;...;ik yi1 þ hi1 ; yi2 þ hi2 ; . . . ; yik þ hik    fi1 ;i2 ;...;ik yi1 ; yi2 ; . . . ; yik : ð48Þ For the sake of notation simplicity, let us rewrite this inequality as:

Dfi1 ;i2 ;...;ik ðxÞ 6 Dfi1 ;i2 ;...;ik ðyÞ;

k1 X

X

fi1 ;i2 ;...;is ðxÞ  f0 ;

ð50Þ

s¼1 i1
one gets:

Dmi1 ;i2 ;...;ik ðyÞ  Dmi1 ;i2 ;...;ik ðxÞ P

k1 X

X

s¼1 i1
Dfi1 ;i2 ;...;is ðyÞ 

k1 X

X

Dfi1 ;i2 ;...;is ðxÞ:



ð51Þ

s¼1 i1
Proof of Corollary 1. If all zi(xi) are non-decreasing, then f is nondecreasing as it is the sum of non-decreasing functions. Conversely, suppose that f is non-decreasing. Thanks to Eq. (6), all zi(xi) = fi + f0, where f0 is a constant. By Theorem 2 if f is non-decreasing, fi is nondecreasing, and all zi are non-decreasing as they differ from the corresponding fi by a constant. h Proof of Corollary 2. Since f is additive, then zi = fi + f0, as in the previous proof. The necessary condition is assured by Theorem 4. The sufficient condition follows from Proposition 1, points (a) and (b). h Proof of Corollary 3. Since f is additive, if it is ultramodular (convex) zi(xi) "i are ultramodular (convex) as well. As shown by? and confirmed in Marinacci and Montrucchio (2005), in the scalar case ultramodularity is equivalent to convexity, provided zi is continuous. It follows that f(x) is convex (ultramodular), since it is the sum of the convex (ultramodular) functions zi(xi). Proof of Corollary 4. If all the zi are increasing and positive, it is immediate to see that f is increasing. Conversely, by and Lemma 2 noting that

mi ðxi Þ Q ¼ zi ðxi Þ; j–i t j where t j ¼ follows. h

R I

tj ;

ð53Þ

j–i

R where t j ¼ I zj ðxj Þdxj "j and tj P 0 by assumption. Ultramodularity of zi(xi) "i follows. The converse statement follows from Proposition 4.4 in Marinacci and Montrucchio (2005) (see Proposition 1, property (c)). h P Proof of Theorem 6. Point 1. Corollary 1 implies that if f ¼ zi ðxi Þ, it is non-decreasing if and only if all the zi are non-decreasing. Now, P in the case of additive independence, uðx; yÞ ¼ ki ui ðti Þ, where P ki > 0.Point 2. Corollary 3, assures that if f is of the form zi ðxi Þ, then f is neg-ultramodular if and only if all zi are neg-ultramodular. (Recall that f is neg-ultramodular if f is ultramodular.) Now, in P the case of additive independence, uðx; yÞ ¼ ki ui ðt i Þ, where ki > 0. Point 3. Follows from Corollary 4. h

ð49Þ

where x and y are here intended as a more synthetic way to represent xi1 ; xi2 ; . . . ; xik and yi1 ; yi2 ; . . . ; yik , respectively. Then, recalling the definition of fi1 ;i2 ;...;ik in Eq. (6), i.e.,

fi1 ;i2 ;...;ik ðxÞ ¼ mi1 ;i2 ;...;ik ðxÞ 

Y

ð52Þ

zj ðxj Þdxj "j and tj P 0 by assumption, the thesis

Proof of Corollary 5. If f is ultramodular, by Lemma 3 the compoR Q nents of the integral decomposition mi ðxi Þ ¼ In1 f ðxÞ k–i dxk are Q ultramodular. If, in addition, f is of the form zi ðxi Þ, then one can write

Proof of Theorem 7. Under  mutual utility independence, Qn   ^0 þ 1 . If k > 0, and u is ultramodular, kuðxÞ þ 1 ¼ i¼1 ku xi ; x then ku(x) + 1 is ultramodular. Hence, by Corollary 5, each of the functions ½kuðxi ; ^ x0 Þ þ 1is ultramodular. By Proposition 1, since k is positive, each uðxi ; ^ x0 Þ is ultramodular. The converse statement is proven as follows. If each uðxi ; ^ x0 Þ is ultramodular and k is positive, kuðxi ; ^ x0 Þ þ 1 is ultramodular for all i. Then, by the second part of Corollary 5, the thesis follows. If k < 0, and u is ultramodular, then ku(x) + 1 is neg-ultramodular. Hence, by Corollary 5, each of the functions ½kuðxi ; ^ x0 Þ þ 1 is neg-ultramodular. By Proposition 1, since k is negative, each uðxi ; ^ x0 Þ is ultramodular. The converse statement is proven as follows. If each uðxi ; ^ x0 Þ is ultramodular and k is negative, 0 ^ kuðxi ; x Þ þ 1 is neg-ultramodular for all i. Then, by the second part of Corollary 5, the thesis follows. h Proof of Theorem 8. Point 1. If vi(xi) are increasing "i, then v(x) is increasing. By theorem 6.11 (Keeney and Raiffa, 1993, p. 330), the utility function corresponding to the preference function in Eq. (29) is either of the multiplicative type

uðxÞ ecv ðxÞ ¼

n Y

ecv i ðxÞ

ð54Þ

i¼1

or proportional to v(x), i.e.,

uðxÞ v ðxÞ:

ð55Þ

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